Bridging the Micro and Macro for Granular Media: A ComputationalMulti-scale Paradigm

J.D. Zhao & N. GuoDepartment of Civil and Environmental Engineering, Hong Kong University of Science and TechnologyClearwater Bay, Kowloon, Hong Kong

ABSTRACT: We present a hierarchical multi-scale framework to model geotechnical problems relevant togranular media. The framework employs a rigorous hierarchical coupling between the finite element method(FEM) and the discrete element method (DEM). The FEM is used to discretise the macroscopic geometric do-main of a boundary value problem into a FEM mesh. A DEM assembly with memory of its loading historyis embedded at each Gauss integration point of the mesh to receive the global deformation from the FEM asinput boundary conditions and is solved for the incremental stress-strain relation at the specific material point toadvance the FEM computation. The hierarchical framework helps to avoid the phenomenological nature of con-ventional continuum approaches of constitutive modelling and meanwhile retains the computational efficiencyof the FEM in solving large-scale boundary value problems (BVPs). By virtue of its hierarchical structure,the predictive power of the proposed method can be further unleashed with proper implementation of parallelcomputing techniques. By corroborating the rich information obtained from the particle level with the macro-scopically observed responses, the framework helps to shed light on a cross-scale understanding of granularmedia. We demonstrate the predictive capability of the proposed framework by simulations of shear localisa-tions in monotonic biaxial compression and cavity inflation in a thick-walled cylinder, as well as liquefactionand cyclic mobility in cyclic simple shear tests.

1 INTRODUCTION

Granular materials are multi-scale by nature. Whensubject to shear, a granular material may exhibitcomplicated macroscopic behaviours that are noto-riously difficult to characterise, such as state depen-dency, strength anisotropy, strain localisation, non-coaxiality, solid-flow phase transition (e.g., liquefac-tion) and critical state (Guo & Zhao 2013b, Zhao& Guo 2013). These macroscopic responses reflectshighly complicated microstructural mechanisms es-tablished and evolved at the granular particle levelduring the loading process. While a granular mediumhas long been treated by continuum mechanics, itbecomes increasingly clear now that a better under-standing can only be achieved with the aid of ef-fective bridging approaches linking the micro to themacro scales of the material. Various homogenisa-tion techniques have been developed in material sci-ences to link different length scales of a material forintegrated characterisation of the material behaviour.They are targeted at designing engineered or new ma-terials with identifiable microstructure to achieve op-timal engineering performance of various purposes.However, several intrinsic properties associated with

granular media exclude the possibility of deriving ma-terial properties and predicting the material responsesdirectly from the particle scale through analytical ho-mogenisation methods as the material science branchdoes. There are two outstanding ones: (1) The lackingof periodic microstructure in a granular material pre-vents a general homogenisation method working ef-fectively, which is due mainly to the great randomnessand heterogeneity within a granular system; (2) Thebehaviour of a granular material is generally state-dependent and loading-path specific. It is thus difficultto identify a once-for-all microstructure from whichthe macroscopic properties of the material can be de-rived via homogenisation. To resolve the bridging is-sue, a computational multi-scale modelling approachappears to be a viable (if not the only) option. In thisstudy, we propose a hierarchical multi-scale frame-work on micro-macro bridging for granular media.The framework employs a rigorous hierarchical cou-pling between the finite element method (FEM) andthe discrete element method (DEM) to solve bound-ary value problems relevant to granular media. Thestudy is in line with a number of previous attempts onthis topic (Meier et al. 2008, Meier et al. 2009, An-drade et al. 2011, Nitka et al. 2011, Guo and Zhao

RVE

Apply deformation

RVE

Stress

Tange

nt o

pera

tor

Strain

Rotation

Macro continuum (BVP)

FEM solver

DEM solver

Material pointor Gauss pointin FEM mesh

Figure 1: A hierarchical multiscale modelling framework forgranular media based on coupled FEM/DEM

2013a).

2 METHODOLOGY AND FORMULATION

2.1 Hierarchical multiscale modelling approach

The multiscale framework employs a rigorous hier-archical coupling between the finite element method(FEM) and the discrete element method (DEM) whichis schematically shown in Fig. 1. To solve a boundaryvalue problem, the macroscopic geometric domain isfirst discretized into a FEM mesh. A DEM assemblyis then embedded at each Gauss integration point ofthe mesh serving as a local Representative VolumeElement (RVE). At each load step, the RVE takes itsmemory of the past loading history as initial condi-tions and receives the global deformation from theFEM at the specific Gauss point as input boundaryconditions. It is solved to derive the local incrementalstress-strain relation (e.g., stress and tangential stiff-ness matrix) required for advancing the global FEMcomputation. To be more specific, the FEM is usedto solve the following equation system for the discre-tised domain:

Ku = f (1)

where K, u and f are the stiffness matrix, the un-known displacement vector at the FEM nodes, and thenodal force vector lumped from the applied bound-ary traction, respectively. Since K is generally de-pendent on the state parameters and loading historyfor a granular medium, linearisation of the solutionand Newton-Raphson iterative method are commonlyrequired. In doing so, the tangent operator Kt needsto properly evaluated:

Kt =∫Ω

BTDB dV (2)

where B is the deformation matrix (i.e. gradient ofthe shape function), and D is the matrix form ofthe rank four tangent operator tensor. During eachNewton-Raphson iteration, both Kt and σ are up-dated to minimise the following residual force R tofind a converged solution

R =∫Ω

BTσ dV − f (3)

Instead of making phenomenological assumptionsfor the instantaneous tangent modulus Dep in Equa-tion (2) to assemble Kt as conventional continuum-based constitutive modelling approaches do (e.g., byassuming an elasto-plastic stiffness matrix in clas-sic plasticity theory), the coupled FEM/DEM multiscale approach determines the two quantities from theembedded discrete element assembly at each Gausspoint. In doing so each DEM packing receives theboundary condition (deformation) by interpolation ofthe FEM solution (displacement). Upon reaching asolution, each DEM is then homogenised to derive thestress and tangent operator at the material point whichare transferred back to the FEM solver to update thesolution. The Cauchy stress tensor is homogenised bythe following expression

σ =1

V

∑Nc

dc ⊗ f c (4)

where V is the total volume of the DEM assembly, Nc

is the number of contacts within the volume, f c anddc are the contact force vector and the branch vectorconnecting the centres of the two contacted particles,respectively.

In deriving the tangent operator, we use the follow-ing bulk elastic modulus homogenised from the DEMassembly as a first coarse estimation of the the tangentoperator:

De =1

V

∑Nc

(knnc⊗dc⊗nc⊗dc+kt t

c⊗dc⊗ tc⊗dc)(5)

where kn and kt are the equivalent normal and tan-gential stiffnesses describing the contact law of theparticles, nc and tc are unit vectors in the outwardnormal and tangential directions of a contact, respec-tively. ‘⊗’ denotes the dyadic product of two vectors.In conduction with an iterative scheme, it works moreefficient in deriving the tangent operator in compar-ison with the alternative perturbation-based method(Meier et al. 2009, Nitka et al. 2011, Guo and Zhao2014).

2.2 Solution procedure

The solution procedure to the hierarchical multi-scalemodeling approach is summarized as follows:

I. Discretise the macro domain by a suitable FEMmesh and embed a DEM assembly with appro-priate initial state at each Gauss point of the meshas a RVE.

II. Apply one global load step imposed by theboundary traction of the FEM domain.

(a) Determine the current tangent operator us-ing Eq. (5) for each RVE .

Figure 2: The calibrated RVE comprised of 400 polydispersespheres with periodic boundary produces isotropic contact nor-mal distribution under isotropic compression.

(b) Assemble the global tangent matrix usingEq. (2) and compute a trial solution of dis-placement u by solving Eq. (1) with FEM.

(c) Interpolate the deformation ∇u at eachGauss point of the FEM mesh and runthe DEM simulation for the correspond-ing RVE by accounting for its initial stateand using ∇u as the DEM boundary con-ditions.

(d) Derive the updated total stress from Eq. (4)for each RVE and use them to evaluate theresidual by Eq. (3) for the FEM domain.

(e) Repeat the above steps from (a) to (d) untilconvergence is reached .

(f) Save the converged solution of each RVE asits new initial state for next step and finishthe current load step

III. Advance to next load step and repeat Step II.

2.3 Benchmark and Calibration

The proposed hierarchical multi-scale approach hasbeen successfully implemented by two open sourcecodes for FEM and DEM (Guo & Zhao 2014). A sim-ple linear force-displacement contact law in conjunc-tion with Coulomb’s friction is employed in the DEMcode to describe the stick-slip inter particle contact.Polydisperse particles with radii ranging from 3 mmto 7 mm (rmean = 5 mm) are adopted to generate eachDEM assembly. Quasi-static solutions are solved ateach load step for each RVE. To calibrate the size ofthe RVE (e.g., particle number), different RVEs withdifferent particle numbers subject to isotropic com-pression have been compared and the resultant con-tact normal distribution is examined. A typical RVEcontaining 400 particles and with periodic boundaryis found to provide largely isotropic contact normaldistribution and meanwhile offer reasonable compu-tational efficiency (see Fig 2), which will be adoptedfor all the simulations in the sequel.

A single-element shear test based on first-orderfour-node quadrilateral element has been used tobenchmark the multi-scale method. All RVEs at thefour Gauss points possess identical initial conditions,

0 1 2 3 4 5 6 7Axial strain, ε11 [%]

50

100

150

200

250

300

Stress[kPa]

σ00

σ11

Pure DEM

Multiscale Approach

0 1 2 3 4 5 6 7Axial strain, ε11 [%]

−4

−3

−2

−1

0

1

Volumetricstrain,εv[%

]

Pure DEM

Multiscale Approach

Figure 3: Comparison of the macroscopic responses of thesingle-element test by the multiscale method and by a pure DEMtest of RVE size.

which leads to a uniform sample for the FEM solu-tion. The global response of the single element test iscompared against that obtained from a pure DEM testbased on the RVE in Fig 3. The global vertical stressσ11 is calculated from the resultant force exerted onthe top boundary and σ00 from the resultant lateralforce. The axial strain ǫ11 and the volumetric strain ǫvare calculated from the overall deformation of the ele-ment. Fig 3 shows that the multi-scale modelling pro-duced nearly identical responses with the pure DEMsimulation for the single element test, which indicatesthat the hierarchical multi-scale modelling approachis able to faithfully reproduce the typical behaviour ofgranular media.

Meanwhile, in our hierarchical multi-scale frame-work, the computation of DEM packing attached toeach Gauss point is independent with one another,which makes it ideal for parallelisation of simula-tions. An effective parallel computing technique hasbeen implemented in our multis-cale code and hasbeen used for all the subsequent simulations with acluster at HKUST. The scalability and efficiency ofthe paralleled code has been found rather satisfactory(Guo & Zhao 2014).

3 DEMONSTRATIVE EXAMPLES

3.1 Monotonic biaxial compression on sand

We first apply the hierarchical multiscale approach topredicting the response of a sand sample subject to

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0 2 4 6 8 10Axial strain, ε11 [%]

100

120

140

160

180

200

220

240

260

280

Axial

stress,σ11[kPa]

Pure DEM

Multiscale Approach

Figure 4: Multiscale modelling of monotonic biaxial compres-sion on sand: (left) model setup; (right) global stress strain re-lationship for the sand sample (calculated from the resultants atthe boundary).

0.0332

0.0322

0.0312

0.0302

0.0292

0.0282

0.0272

0.0262

Void ratio

0.174

0.174

0.174

0.174

0.174

0.173

0.173

0.173

(a) ε11 = 1.6% (peak)

0.680

0.584

0.488

0.392

0.296

0.200

0.104

0.0076

Void ratio

0.250

0.239

0.228

0.217

0.206

0.195

0.184

0.174

(b) ε11 = 10% (final)

Figure 5: Contours of the accumulated deviatoric strain and voidratio showing strain localisation in biaxial compression of sand.

monotonic biaxial compression. The model setup andboundary conditions are shown in Fig. 4 (left) wherethe FEM mesh consists of four-node linear quadrilat-eral elements. Fig. 4 (right) presents the global stress-strain response measured from the boundary reactionforces and displacements. A comparison case is alsopresented for the pure DEM simulation of the biax-ial shear on a RVE size sample. Notably, the globalresponse by the multiscale modelling approach bearsgreat similarity to the RVE response, in particular forthe pre-peak stress stage where the material behaviouris relatively elastic. While the post-peak response ofthe DEM test shows moderate fluctuations, the multi-scale model results are relatively smooth.

Our multi-scale simulation captures the phe-nomenon of strain localisation that is commonlyfound in laboratory biaxial shear tests (see Fig. 5).This is interesting by the following reason. Our bi-axial shear simulation has been set up with smoothand symmetric boundary conditions and loading. Theinitial states for all RVE packings in the FEM meshare identical and hence the entire sample is homoge-neous too. Under such symmetric/homogeneous con-ditions, conventional continuum-model-based FEMapproaches are generally unable to capture the strainlocalisation unless certain artificial imperfection(s) orrandom distributed local properties are added to thesample to break the symmetry and trigger strain lo-calisation. In our multi-scale modelling of the prob-lem, though the RVEs are identical and homogeneousin the FEM mesh, there is small but observable ini-tial anisotropy present in the initial RVE packing (seeFig. 2), which may serve as the symmetry breaker to

Initial state of all RVEs Final state of N°51

Final state of N°422Final state of N°256

>500

400

300

200

100

0

[N]

(a)

45°135°

225° 315°

0°

90°

180°

270°

N°51

45°135°

225° 315°

0°

90°

180°

270°

N°256

(b)

Figure 6: (a) The structure and force chains of the local initialand deformed RVEs and (b) the corresponding distributions ofthe contact normals in N51 and N256. The smooth red curvesin (b) are the second-order Fourier approximations.

trigger the occurrence in our mutliscale modelling.Similar opinion has been discussed by Gao & Zhao(2013) based on continuum modelling. The localisedregion observed from Fig. 5 concentrates in the mostdilative portion of the sample with large void ratio,which is consistent with experimental observations.The localised band of void ratio is found generallywider than that of the deviatoric strain.

A major advantage of the multi-scale frameworklies in the rich micro-scale information it can pro-vide when solving an engineering scale problem. Thismay greatly facilitate a better correlation and under-standing of the macroscopic observations. Shown inFig. 6 is a comparison of the contact force networkfor the RVE packings at the three Gauss points indi-cated in Fig. 4 at the initial and final loading states.All RVEs have the same initial isotropic conditionwithout apparent preferably orientated strong forcechains. While the DEM packing at N51 does notexperience much deformation, several distinct strongforce chains are observed aligning parallel to the ver-tical shear direction. In contrast, the RVE at N256experiences severe shear deformation which resultsin a highly heterogeneous internal structure and moreconcentrated penetrating force chains aligning to thevertical. The packing at N422 also deforms notice-ably, but it becomes rather dilute due to remarkablevolumetric expansion. The deformation gradients atN256 and N422 are consistent with the global shearband inclination. An inspection of the contact nor-mal distribution (shown in Fig. 6 ) further confirmsthat the major principal directions for both N51 andN256 are close to the vertical shear direction.

3.2 Cyclic simple shear test

The multiscale modelling approach has also been ap-plied to simulating the hysteresis behaviour of sandsubject to cyclic load. Shown in Fig. 7 is a FEM meshand boundary conditions for a sand sample subject tomaximum shear stress controlled cyclic simple shear.The initial packing is chosen to be relatively loosefor all RVEs (different than the other examples where

Rough Loading Platen

Periodicboundary

50

mm

100 mm

N°390

Figure 7: Discretization (15× 10 elements) and boundary con-dition of the specimen for cyclic simple shear. The N390 Gausspoint will be used for local analyses.

−5 −4 −3 −2 −1 0 1 2Shear strain, γ01 [%]

−40

−30

−20

−10

0

10

20

30

40

Shearstress,σ01[kPa]

(a) Shear stress-strain relation

40 50 60 70 80 90 100 110Normal stress, σ11 [kPa]

−40

−30

−20

−10

0

10

20

30

40

Shearstress,σ01[kPa]

(b) Normal-shear stress relation

Figure 8: Global responses of the cyclic simple shear test undermaximum shear stress control |σ01| = 30kPa.

Void ratio

0.209

0.209

0.208

0.208

0.207

(a)

300

240

180

120

60

0

[N]

360

(b)

Figure 9: (a) Contour of void ratio at the final state after the max-imum shear stress controlled cyclic loading and (b) the structureand the force chains in RVE at N390.

(a)

0

50

100

150

200

250

300

0 3 6 9 12 15

σc:

kP

a

uc: mm

(b)

Figure 10: Multiscale simulation of cavity inflation of a thick-walled hollow cylinder in sand: (a) mesh; (b) pressure-inflationcurve.

dense packing is used). The global stress-strain re-sponse and loading path are shown in Fig. 8, indicat-ing a typical hysteresis behaviour of sand observedin laboratory tests. The contour of the void ratio inFig. 9 indicates that the deformation in the sampleis relatively uniform. The RVE at N390 shows sev-eral obvious strong force chains aligning along the di-agonal direction of the packing. As the accumulatedshear strain becomes larger, it is evident that cyclicmobility occurs for this sample. Not presenting heredue to length limitation, we have also investigated acomparison case where the same sample is subjectto constant-volume maximum shear strain controlledcyclic shear wherein we found liquefaction occurs. Atthe liquified material points, the contact network ofthe RVE become too week to form effective percolat-ing force chains to sustain the external shear.

3.3 Cavity inflation in thick-walled hollow cylinder

The multi-scale approach has also been applied tomodelling the cavity inflation in thick-walled cylin-der as treated experimentally by Alsiny et al. (1992).Thick-walled hollow cylinder inflation experimentsare commonly performed towards a better understandthe soil behaviour under passive loading conditionswith simultaneous extension in a plane perpendicularto the loading direction. A quarter of a whole thick-walled hollow cylinder is simulated, with an identicalgeometry of that in Alsiny et al. (1992) for the cav-ity and outer surface of the cylinder: rc = 15 mm andro = 150 mm. Due to symmetry, the displacement of

the left boundary is fixed and the vertical displace-ment of the bottom boundary is fixed (see Fig. 10a).Eight-node elements were used in the study. The outerboundary is prescribed by a constant pressure p0 =100 kPa. Different than the inflation pressure bound-ary in Alsiny et al. (1992), a Neumann boundarywith gradually increased displacement uc is appliedto the inner surface until uc = 10 mm. Fig. 10b de-picts the inflation-pressure response (the differentialcavity pressure is defined by the pressure differenceon the inner and outer cylinder walls), which showsa clear softening portion of the curve. Our simulationshows that the occurrence of shear localisation in thecylinder was initiated shortly beyond the peak infla-tion pressure, which is consistent with the experimen-tal finding by Alsiny et al. (1992). Since our simula-tion uses Neumann boundary at the cavity surface, nodiffuse deformation mode has been detected.

Fig. 11 presents the localised distributions of shearstrain, void ratio and average rotation (obtained fromthe RVE) within the cylinder at the end of the infla-tion. The three contours show apparently good cor-relations. Within the shear bands the soil is severelysheared, highly dilated (initial void ratio is 0.177)and experiences considerable rotation of soil parti-cles. The shear bands originated from the cavity sur-face form cross-shaped patterns which have been ob-served in laboratory tests. The soil particles withinthe bands of different orientation have been rotatedin totally different direction. Other than those close tothe cavity, there are two less developed, notable shearbands touching the outer surface of the cylinder. Theyare however not originated directly from the cavitysurface, but from the two fixed boundaries. They ap-pear to be reflections of the two major shear bandsinitiated from the cavity surface. Further investigationbased on a full cylinder rather than its quarter will becarried out to see if this is the case. The shear bandorientation will also be analysed based on more re-fined model of the problem in the future.

4 CONCLUSIONS

A hierarchical multi-scale framework has been de-veloped to bridge the micro and macro behavioursof granular media for practical modelling of engi-neering scale problems. Based on a rigorous cou-pling between FEM and DEM, it circumvents thephenomenological assumptions commonly requiredin continuum constitutive modelling and meanwhileretains a good predictive capability on solving prac-tical problems which purely micromechanics-basedapproaches are inherently restrained to solve. Theframework has been benchmarked, calibrated and en-hanced with parallel computing techniques. Its pre-dictive capability has been demonstrated with threeillustrative examples, including monotonic biaxialcompression test, cyclic simple shear tests and cavityinflation in thick-walled hollow cylinder.

Shear strain

1.81

0.008

1.6

1.2

0.8

0.4

(a)

Void ratio

0.236

0.177

0.22

0.18

0.2

(b)

0.334

-0.388

0.2

0

-0.2

Avg. rot.

(c)

Figure 11: Localised distributions for (a) shear strain, (b) voidratio and (c) average rotation in the cylinder

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